**Variance Formula (Table of Contents)**

## What is a Variance Formula?

The term “variance” refers to the extent of dispersion of the data points of a data set from its mean, which is computed as the average of the squared deviation of each data point from the population mean. The formula for a variance can be derived by summing up the squared deviation of each data point and then dividing the result by the total number of data points in the data set. Mathematically, it is represented as,

**σ**

^{2}= ∑ (X_{i}– μ)^{2}/ Nwhere,

**X**= i_{i}^{th}data point in the data set**μ**= Population mean**N**= Number of data points in the population

**Examples of Variance Formula (With Excel Template)**

Let’s take an example to understand the calculation of the Variance in a better manner.

#### Variance Formula – Example #1

**Let us take the example of a classroom with 5 students. The class had a medical check-up wherein they were weighed, and the following data was captured. Calculate the variance of the data set based on the given information. **

**Solution: **

Population Mean is calculated as:

- Population Mean = (30 kgs + 33 kgs + 39 kgs + 29 kgs + 34 kgs) / 5
- Population Mean =
**33 kgs**

Now, we need to calculate the deviation, i.e. difference between the data points and the mean value.

Similarly, calculate for all values of the data set.

Now, let us calculate the squared deviations of each data point as shown below,

Variance is calculated using the formula given below

**σ ^{2} = ∑ (X_{i} – μ)^{2} / N**

- σ
^{2 }= (9 + 0 + 36 + 16 + 1) / 5 - σ
^{2 }=**12.4**

Therefore, the variance of the data set is **12.4**.

#### Variance Formula – Example #2

**Let us take the example of a start-up company that comprises 8 people. The age of all the members is given. Calculate the variance of the data set based on the given information. **

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**Solution:**

Population Mean is calculated as:

- Population Mean = (23 years + 32 years + 27 years + 37 years + 35 years + 25 years + 29 years + 40 years) / 8
- Population Mean =
**31 years**

Now, we need to calculate the deviation, i.e. difference between the data points and the mean value.

Similarly, calculate for all values of the data set.

Now, let us calculate the squared deviations of each data point as shown below,

Variance is calculated using the formula given below

**σ ^{2} = ∑ (X_{i} – μ)^{2} / N**

- σ
^{2}= (64 + 1 + 16 + 36 + 16 + 36 + 4 + 81) / 8 - σ
^{2}=**31.75**

Therefore, the variance of the data set is **31.75**.

### Explanation

The formula for a variance can be derived by using the following steps:

**Step 1: **Firstly, create a population comprising a large number of data points. Xi will denote these data points.

**Step 2:** Next, calculate the number of data points in the population which is denoted by N.

**Step 3:** Next, calculate the population means by adding up all the data points and then dividing the result by the total number of data points (step 2) in the population. The population means is denoted by μ.

**μ = X _{1} + X_{2} + X_{3} + X_{4} + X_{5 }/ N**

or

**μ = ∑ X _{i }/ N**

**Step 4: **Next, subtract the population mean from each of the data points of the population to determine the deviation of each of the data points from the mean, i.e. (X_{1} – μ) is the deviation for the 1^{st} data point, while (X_{2} – μ) is for the 2^{nd} data point, etc.

**Step 5: **Next, determine the square of all the respective deviations calculated in step 4, i.e. (X_{i} – μ)^{2}.

**Step 6:** Next, sum up all the of the respective squared deviations calculated in step 5, i.e. (X_{1} – μ)^{2} + (X_{2} – μ)^{2 }+ (X_{3} – μ)^{2} + …… + (X_{n} – μ)^{2} or ∑ (X_{i} – μ)^{2}.

**Step 7:** Finally, the formula for a variance can be derived by dividing the sum of the squared deviations calculated in step 6 by the total number of data points in the population (step 2), as shown below.

**σ ^{2} = ∑ (X_{i} – μ)^{2} / N**

### Relevance and Uses of Variance Formula

From the perspective of a statistician, a variance is a very important concept to understand as it is often used in probability distribution to measure the variability (volatility) of the data set vis-à-vis its mean. The volatility serves as a measure of risk, and as such, the variance is helpful in assessing an investor’s portfolio risk. A zero variance is signifying that all variables in the data set are identical. On the other hand, a higher variance can be indicative of the fact that all the variables in the data set are far-off from the mean, while a lower variance signifies exactly the opposite. Please keep in mind that variance can never be a negative number.

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